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Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Define entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ by $$H(\mathcal{A},m)=-\sum_{i=0}^mp_i\log_2p_i.$$

If $k>0$, entropy of $\mathcal{B}=\{ka_i\}_{i=1}^m$ $$H(\mathcal{B},m)=H(\mathcal{A},m).$$

(Scaling fixes probability distribution which leads to same entropy).

Is there another natural transformation such that entropies remain fixed?

Would it be possible to obtain another transformation when considering special sequences?

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    $\begingroup$ Your example not only leaves H fixed, but also p. $\endgroup$
    – guest
    Commented Jan 3, 2015 at 21:49
  • $\begingroup$ I know that fixes $H(\mathcal{B},m)$. Could there be another transformation? $\endgroup$
    – Turbo
    Commented Jan 3, 2015 at 22:30
  • $\begingroup$ 1. Why is it important that they are integers? (Any reason you cannot operate on sorted $p_i$?) 2. It is important that it is Shannon, not some other entropy, e.g. a function of $\sum p_i^2$? $\endgroup$
    – Piotr Migdal
    Commented Jan 11, 2015 at 15:32
  • $\begingroup$ Only entropy I am familiar with is Shannon's. Please feel free to write other insights if possible. Reals also would work. Integer sequences look less messy. Do reals help to solve? $\endgroup$
    – Turbo
    Commented Jan 11, 2015 at 16:13
  • $\begingroup$ "insights" are too open-ended for MO. Restricting to integers make it more complicated (e.g. you loose continuity, you make problem description longer, ...). $\endgroup$
    – Piotr Migdal
    Commented Jan 27, 2015 at 12:09

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